Boundary values of the resolvent of Schrödinger hamiltonians with potentials of order zero

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March 2013, 33(3): 1061-1076. doi: 10.3934/dcds.2013.33.1061

Boundary values of the resolvent of Schrödinger hamiltonians with potentials of order zero

J. Cruz-Sampedro ^1,

1.	Departamento de Ciencias Básicas, UAM-A, Av. San Pablo 180, Col. Reynosa, México D.F. 02200, Mexico

Received April 2011 Revised October 2011 Published October 2012

Abstract
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Let $H=-\Delta +V$ be a Schrödinger hamiltonian acting on $L^2(\mathbb{R}^n)$, $n\geq 2$, where $V$ a potential of order zero plus a short-range perturbation. In this work we investigate the behavior of the resolvent $R(z)=(H-z)^{-1}$ of $H$ as Im$\,z \downarrow 0$, at high energies and in the framework of Besov spaces $B(\mathbb{R}^n)$. For $\lambda_0>0$ sufficiently large and $\lambda\geq\lambda_0$, we show that there exists a linear operator $R(\lambda+i0)$ such that $R(\lambda+i\epsilon)$ converges to $R(\lambda+i0)$ as $\epsilon\downarrow 0$, strongly in $\mathcal{L}(L^{2, s}(\mathbb{R}^n),L^{2,-s}(\mathbb{R}^n))$, $s>1/2$, and weakly in $\mathcal{L}(B(\mathbb{R}^n),B^*(\mathbb{R}^n))$. We achieve this through a Mourre-estimate strategy.

Keywords: order-zero potentials, Schrödinger hamiltonian, Besov spaces..

Mathematics Subject Classification: Primary: 35P25, 81U99, 47A40; Secondary: 46C9.

Citation: J. Cruz-Sampedro. Boundary values of the resolvent of Schrödinger hamiltonians with potentials of order zero. Discrete and Continuous Dynamical Systems, 2013, 33 (3) : 1061-1076. doi: 10.3934/dcds.2013.33.1061

References:

[1]	S. Agmon, Spectral properties of Schrödinger operators and scattering theory, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 2 (1975), 151-218.
[2]	S. Agmon, J. Cruz and I. Herbst, Generalized Fourier transform for Schrödinger operators with potentials of order zero, Journal of Functional Analysis, 167 (1999), 345-369. doi: 10.1006/jfan.1999.3432.
[3]	S. Agmon and L. Hörmander, Asymptotic properties of solutions of differential equations with simple characteristics, Journal D'Analyse Mathématique, 30 (1976), 1–-38.
[4]	G. Barles, On eikonal equations associated with Schrödinger operators with nonspherical radiation conditions, Commun. in Partial Differential Equations, 12 (1987), 263-283.
[5]	J. Cruz-Sampedro, The eikonal equation and a class of Schrödinger-like operators, Submitted 2011.
[6]	J. Dereziński and C. Gérard, "Scattering Theory of Classical and Quantum $N$-Particle Systems," Springer, 1997.
[7]	W. Jäger, Über das Dirichletsche Auβenraumproblem fü die Schwingungsgleichung, Math. Zeitschr, 95 (1967), 299-323. doi: 10.1007/BF01111082.
[8]	A. Jensen and P. Perry, Commutator methods and Besov space estimates for Schrödinger operators, J. Operator Theory, 14 (1985), 181-188.
[9]	D. Jerison and C. Kenig, Unique continuation and absence of positive eigenvalues for Schrödinger operators, Ann. Math., 121 (1985), 463-494. doi: 10.2307/1971205.
[10]	E. Mourre, Absence of singular continuous spectrum for certain self-adjoint operators, Commun. Math. Phys., 78 (1980/81), 391-408. doi: 10.1007/BF01942331.
[11]	P. Perry, I. Sigal and B. Simon, Spectral analysis of $N$-body Schrödinger operators, Ann. Math., 114 (1981), 519-567. doi: 10.2307/1971301.
[12]	M. Reed and B. Simon, "Methods of Modern Mathematical Physics, III Sacattering Theory," New York, Academic Press, 1979.
[13]	M. Reed and B. Simon, "Methods of Modern Mathematical Physics, IV Analysis of Operators," Academic Press, 1978.
[14]	Y. Saitō, Schrödinger operators with a nonspherical radiation condition, Pacific Journal of Mathematics, 126 (1987), 331-359.
[15]	I. Sigal, "Scattering Theory for Many-Body Quantum Mechanical Systems," Lecture Notes in Mathematics 1011, Springer Verlag 1983.
[16]	G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, (French) Ann. Inst. Fourier (Grenoble), 15 (1965), 189-258.

show all references

References:

[1]	S. Agmon, Spectral properties of Schrödinger operators and scattering theory, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 2 (1975), 151-218.
[2]	S. Agmon, J. Cruz and I. Herbst, Generalized Fourier transform for Schrödinger operators with potentials of order zero, Journal of Functional Analysis, 167 (1999), 345-369. doi: 10.1006/jfan.1999.3432.
[3]	S. Agmon and L. Hörmander, Asymptotic properties of solutions of differential equations with simple characteristics, Journal D'Analyse Mathématique, 30 (1976), 1–-38.
[4]	G. Barles, On eikonal equations associated with Schrödinger operators with nonspherical radiation conditions, Commun. in Partial Differential Equations, 12 (1987), 263-283.
[5]	J. Cruz-Sampedro, The eikonal equation and a class of Schrödinger-like operators, Submitted 2011.
[6]	J. Dereziński and C. Gérard, "Scattering Theory of Classical and Quantum $N$-Particle Systems," Springer, 1997.
[7]	W. Jäger, Über das Dirichletsche Auβenraumproblem fü die Schwingungsgleichung, Math. Zeitschr, 95 (1967), 299-323. doi: 10.1007/BF01111082.
[8]	A. Jensen and P. Perry, Commutator methods and Besov space estimates for Schrödinger operators, J. Operator Theory, 14 (1985), 181-188.
[9]	D. Jerison and C. Kenig, Unique continuation and absence of positive eigenvalues for Schrödinger operators, Ann. Math., 121 (1985), 463-494. doi: 10.2307/1971205.
[10]	E. Mourre, Absence of singular continuous spectrum for certain self-adjoint operators, Commun. Math. Phys., 78 (1980/81), 391-408. doi: 10.1007/BF01942331.
[11]	P. Perry, I. Sigal and B. Simon, Spectral analysis of $N$-body Schrödinger operators, Ann. Math., 114 (1981), 519-567. doi: 10.2307/1971301.
[12]	M. Reed and B. Simon, "Methods of Modern Mathematical Physics, III Sacattering Theory," New York, Academic Press, 1979.
[13]	M. Reed and B. Simon, "Methods of Modern Mathematical Physics, IV Analysis of Operators," Academic Press, 1978.
[14]	Y. Saitō, Schrödinger operators with a nonspherical radiation condition, Pacific Journal of Mathematics, 126 (1987), 331-359.
[15]	I. Sigal, "Scattering Theory for Many-Body Quantum Mechanical Systems," Lecture Notes in Mathematics 1011, Springer Verlag 1983.
[16]	G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, (French) Ann. Inst. Fourier (Grenoble), 15 (1965), 189-258.

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